is well defined only if the Euler characteristic of \(M\)  is zero.
The energy \(F\)  is interesting both from the geometric and the modellistic point of view.
On the one hand, the energy \(F\)  is usually called Total Bending (see [1]}) and (quantitatively) measures how a globally defined unit-norm vector field \(u\) , if it exists, fails to be a parallel vector field (namely a vector field \(u\)  for which \(D_v u =0\)  for any \(v\) ).
On the other hand, the energy \(F\)  is related, for instance, to the modelling of nematic shells, namely a rigid particle coated by a thin film of nematic liquid crystal (see [2]}, [3]}, [4]}, [5]}, [6]}, [7]}) when one neglects the effects of the extrinsic geometry of the substrate on the elastic energy. We refer to [8]}, [9]} and [10]} for models including extrinsic effects.
